<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
BDDC	B-Algorithm
(	O
balancing	B-Algorithm
domain	I-Algorithm
decomposition	I-Algorithm
by	I-Algorithm
constraints	I-Algorithm
)	O
is	O
a	O
domain	B-Algorithm
decomposition	I-Algorithm
method	I-Algorithm
for	O
solving	O
large	O
symmetric	B-Algorithm
,	O
positive	B-Algorithm
definite	I-Algorithm
systems	O
of	O
linear	O
equations	O
that	O
arise	O
from	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
BDDC	B-Algorithm
is	O
used	O
as	O
a	O
preconditioner	O
to	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
specific	O
version	O
of	O
BDDC	B-Algorithm
is	O
characterized	O
by	O
the	O
choice	O
of	O
coarse	O
degrees	O
of	O
freedom	O
,	O
which	O
can	O
be	O
values	O
at	O
the	O
corners	O
of	O
the	O
subdomains	O
,	O
or	O
averages	O
over	O
the	O
edges	O
or	O
the	O
faces	O
of	O
the	O
interface	O
between	O
the	O
subdomains	O
.	O
</s>
<s>
One	O
application	O
of	O
the	O
BDDC	B-Algorithm
preconditioner	O
then	O
combines	O
the	O
solution	O
of	O
local	O
problems	O
on	O
each	O
subdomains	O
with	O
the	O
solution	O
of	O
a	O
global	O
coarse	B-Algorithm
problem	I-Algorithm
with	O
the	O
coarse	O
degrees	O
of	O
freedom	O
as	O
the	O
unknowns	O
.	O
</s>
<s>
The	O
local	O
problems	O
on	O
different	O
subdomains	O
are	O
completely	O
independent	O
of	O
each	O
other	O
,	O
so	O
the	O
method	O
is	O
suitable	O
for	O
parallel	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
With	O
a	O
proper	O
choice	O
of	O
the	O
coarse	O
degrees	O
of	O
freedom	O
(	O
corners	O
in	O
2D	O
,	O
corners	O
plus	O
edges	O
or	O
corners	O
plus	O
faces	O
in	O
3D	O
)	O
and	O
with	O
regular	O
subdomain	O
shapes	O
,	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
method	O
is	O
bounded	O
when	O
increasing	O
the	O
number	O
of	O
subdomains	O
,	O
and	O
it	O
grows	O
only	O
very	O
slowly	O
with	O
the	O
number	O
of	O
elements	O
per	O
subdomain	O
.	O
</s>
<s>
BDDC	B-Algorithm
was	O
introduced	O
by	O
different	O
authors	O
and	O
different	O
approaches	O
at	O
about	O
the	O
same	O
time	O
,	O
i.e.	O
,	O
by	O
Cros	O
,	O
Dohrmann	O
,	O
and	O
Fragakis	O
and	O
Papadrakakis	O
,	O
as	O
a	O
primal	O
alternative	O
to	O
the	O
FETI-DP	B-Algorithm
domain	B-Algorithm
decomposition	I-Algorithm
method	I-Algorithm
by	O
Farhat	O
et	O
al	O
.	O
</s>
<s>
See	O
for	O
a	O
proof	O
that	O
these	O
are	O
all	O
actually	O
the	O
same	O
method	O
as	O
BDDC	B-Algorithm
.	O
</s>
<s>
The	O
name	O
of	O
the	O
method	O
was	O
coined	O
by	O
Mandel	O
and	O
Dohrmann	O
,	O
because	O
it	O
can	O
be	O
understood	O
as	O
further	O
development	O
of	O
the	O
BDD	O
(	O
balancing	B-Algorithm
domain	I-Algorithm
decomposition	I-Algorithm
)	O
method	O
.	O
</s>
<s>
Mandel	O
,	O
Dohrmann	O
,	O
and	O
Tezaur	O
proved	O
that	O
the	O
eigenvalues	O
of	O
BDDC	B-Algorithm
and	O
FETI-DP	B-Algorithm
are	O
identical	O
,	O
except	O
for	O
the	O
eigenvalue	O
equal	O
to	O
one	O
,	O
which	O
may	O
be	O
present	O
in	O
BDDC	B-Algorithm
but	O
not	O
for	O
FETI-DP	B-Algorithm
,	O
and	O
thus	O
their	O
number	O
of	O
iterations	O
is	O
practically	O
the	O
same	O
.	O
</s>
<s>
The	O
coarse	B-Algorithm
space	I-Algorithm
of	O
BDDC	B-Algorithm
consists	O
of	O
energy	O
minimal	O
functions	O
with	O
the	O
given	O
values	O
of	O
the	O
coarse	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
This	O
is	O
the	O
same	O
coarse	B-Algorithm
space	I-Algorithm
as	O
used	O
for	O
corners	O
in	O
a	O
version	O
of	O
BDD	O
for	O
plates	O
and	O
shells	O
.	O
</s>
<s>
The	O
difference	O
is	O
that	O
in	O
BDDC	B-Algorithm
,	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
used	O
in	O
an	O
additive	O
fashion	O
,	O
while	O
in	O
BDD	O
,	O
it	O
is	O
used	O
a	O
multiplicatively	O
.	O
</s>
<s>
The	O
BDDC	B-Algorithm
method	O
is	O
often	O
used	O
to	O
solve	O
problems	O
from	O
linear	O
elasticity	O
,	O
and	O
it	O
can	O
be	O
perhaps	O
best	O
explained	O
in	O
terms	O
of	O
the	O
deformation	O
of	O
an	O
elastic	O
structure	O
.	O
</s>
<s>
After	O
applying	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
we	O
obtain	O
a	O
system	O
of	O
linear	O
algebraic	O
equations	O
,	O
where	O
the	O
unknowns	O
are	O
the	O
displacements	O
at	O
the	O
nodes	O
of	O
the	O
elements	O
and	O
the	O
right-hand	O
side	O
comes	O
from	O
the	O
forces	O
(	O
and	O
from	O
nonzero	O
prescribed	O
displacements	O
on	O
the	O
boundary	O
,	O
but	O
,	O
for	O
simplicity	O
,	O
assume	O
that	O
these	O
are	O
zero	O
)	O
.	O
</s>
<s>
The	O
first	O
step	O
in	O
the	O
BDDC	B-Algorithm
method	O
is	O
the	O
interior	O
correction	O
,	O
which	O
consists	O
of	O
finding	O
the	O
deformation	O
of	O
each	O
subdomain	O
separately	O
given	O
the	O
forces	O
applied	O
to	O
the	O
subdomain	O
except	O
at	O
the	O
interface	O
of	O
the	O
subdomain	O
with	O
its	O
neighbors	O
.	O
</s>
<s>
Thus	O
,	O
the	O
interface	O
forces	O
are	O
averaged	O
and	O
the	O
coarse	O
solution	O
is	O
found	O
by	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
gives	O
the	O
value	O
of	O
the	O
output	O
of	O
BDDC	B-Algorithm
on	O
the	O
interfaces	O
between	O
the	O
subdomains	O
.	O
</s>
<s>
The	O
values	O
of	O
the	O
output	O
of	O
BDDC	B-Algorithm
in	O
the	O
interior	O
of	O
the	O
subdomains	O
are	O
then	O
obtained	O
by	O
repeating	O
the	O
interior	O
correction	O
.	O
</s>
<s>
Then	O
the	O
forces	O
inside	O
the	O
subdomains	O
stay	O
zero	O
during	O
the	O
conjugate	B-Algorithm
gradients	I-Algorithm
iterations	O
,	O
and	O
so	O
the	O
first	O
interior	O
correction	O
in	O
each	O
application	O
of	O
BDDC	B-Algorithm
can	O
be	O
omitted	O
.	O
</s>
